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Register a new userReduction of branching graphs supporting continuous time return quantum walks
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We demonstrate that continuous time quantum walks on several types of branching graphs, including graphs with loops, are identical to quantum walks on simpler linear chain graphs. We also show graph types for which such equivalence does not exist. Several instructive examples are discussed, and a general approach to analyze more complex branching graphs is formulated. It is further illustrated with a return quantum walk solution for a cube graph with adjustable complex hopping amplitudes.
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We address continuous-time quantum walks on graphs in the presence of time- and space-dependent noise. Noise is modeled as generalized dynamical percolation, i.e. classical time-dependent fluctuations affecting the tunneling amplitudes of the walker. In order to illustrate the general features of the model, we review recent results on two paradigmatic examples: the dynamics of quantum walks on the line and the effects of noise on the performances of quantum spatial search on the complete and the star graph. We also discuss future perspectives, including extension to many-particle quantum walk, to noise model for on-site energies and to the analysis of different noise spectra. Finally, we address the use of quantum walks as a quantum probe to characterize defects and perturbations occurring in complex, classical and quantum, networks.
Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and on the specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but in general they are uncorrelated.
We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples. Let $X$ be a graph with adjacency matrix $A$ and consider quantum walks on the vertex set $V(X)$ governed by the continuous time-dependent unitary transition operator $U(t)= exp(itA)$. For $S,Tsubseteq V(X)$, we says $X$ admits group state transfer from $S$ to $T$ at time $tau$ if the submatrix of $U(tau)$ obtained by restricting to columns in $S$ and rows not in $T$ is the all-zero matrix. As a generalization of perfect state transfer, fractional revival and periodicity, group state transfer satisfies natural monotonicity and transitivity properties. Yet non-trivial group state transfer is still rare; using a compactness argument, we prove that bijective group state transfer (the optimal case where $|S|=|T|$) is absent for almost all $t$. Focusing on this bijective case, we obtain a structure theorem, prove that bijective group state transfer is monogamous, and study the relationship between the projections of $S$ and $T$ into each eigenspace of the graph. Group state transfer is obviously preserved by graph automorphisms and this gives us information about the relationship between the setwise stabilizer of $Ssubseteq V(X)$ and the stabilizers of naturally defined subsets obtained by spreading $S$ out over time and crudely reversing this process. These operations are sufficiently well-behaved to give us a topology on $V(X)$ which is likely to be simply the topology of subsets for which bijective group state transfer occurs at that time. We illustrate non-trivial group state transfer in bipartite graphs with integer eigenvalues, in joins of graphs, and in symmetric double stars. The Cartesian product allows us to build new examples from old ones.
In this paper we present a model exhibiting a new type of continuous-time quantum walk (as a quantum mechanical transport process) on networks, which is described by a non-Hermitian Hamiltonian possessing a real spectrum. We call it pseudo-Hermitian continuous-time quantum walk. We introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system. We observe that the probability distribution on certain vertices increases compared to that of the Hermitian case. This formalism makes the transport process faster and can be useful for search algorithms.
We propose a definition for the Polya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, higher dimensional integer lattices and a number of other graphs and calculate their Polya number. For the timing of the measurements a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.
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